To solve the equation
[tex]\[
\frac{x+3}{x+4}+\frac{3}{x}=\frac{3}{x^2+4x}
\][/tex]
we need to find the values of [tex]\(x\)[/tex] that satisfy this equation.
Let's break this down step-by-step:
1. Combine the denominators on the left-hand side:
The given equation is:
[tex]\[
\frac{x+3}{x+4} + \frac{3}{x} = \frac{3}{x^2 + 4x}
\][/tex]
2. Express all terms with a common denominator:
The left-hand side can be simplified:
[tex]\[
\frac{(x+3)x + 3(x+4)}{(x+4)x} = \frac{3}{x^2 + 4x}
\][/tex]
Combine the fractions on the left:
[tex]\[
\frac{x^2 + 3x + 3x + 12}{x(x+4)} = \frac{3}{x^2 + 4x}
\][/tex]
Simplify the numerator:
[tex]\[
\frac{x^2 + 6x + 12}{x^2 + 4x} = \frac{3}{x^2 + 4x}
\][/tex]
3. Cross-multiply to eliminate the denominators:
Equating the numerators:
[tex]\[
x^2 + 6x + 12 = 3
\][/tex]
4. Solving the quadratic equation:
[tex]\[
x^2 + 6x + 12 - 3 = 0
\][/tex]
[tex]\[
x^2 + 6x + 9 = 0
\][/tex]
This can be factored as:
[tex]\[
(x + 3)^2 = 0
\][/tex]
Taking the square root of both sides:
[tex]\[
x + 3 = 0
\][/tex]
5. Solving for [tex]\(x\)[/tex]:
[tex]\[
x = -3
\][/tex]
Therefore, the only solution to the equation is [tex]\( x = -3 \)[/tex].
Considering the given answer choices:
- Correct answer: [tex]\(-3\)[/tex]
- 0, 1, and 3 are not solutions to the equation.
So, the correct answer is:
[tex]\[ \boxed{-3} \][/tex]
